= Weak transport solution under characteristic coordinates
{title2=$u(t,X(t,a))=u_0(a)$}
For a complete $C^1$ velocity flow, the bounded <weak solution> of $u_t+F\cdot\nabla u=0$ is its initial value pulled back by the inverse flow. Its weak identity is $\int u(\varphi_t+\operatorname{div}(F\varphi))+\int u_0\varphi(0)=0$; the divergence term is essential. Changing to flow coordinates gives $\int w\partial_t(J\zeta)+\int u_0\zeta(0)=0$. Approximation in the spatial parameter permits $\zeta=\eta/J$, since $J$ and $J_t$ are continuous and $J>0$ on compact sets. Hence $\partial_tw=0$ distributionally, proving uniqueness without requiring second <derivatives> of $F$.
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